Business Mathematics and Statistics by Asim Kumar Manna (z-lib.org)

CHAPTER

8

Index Numbers

SYLLABUS

Meaning and Types of Index Numbers, Problems of Constructing Index
Numbers, Construction of Price and Quantity Indices, Test of Adequacy,
Errors in Index Numbers, Chain Base Index Numbers; Base Shifting,
Splicing, Deflating, Consumer Price Index and its Uses

THEMATIC FOCUS

8.1 Introduction
8.1.1 Meaning of Index Number

8.2 Types of Index Numbers
8.2.1 Price Index Numbers
8.2.2 Quantity Index Numbers
8.2.3 Value Index Numbers

8.3 Characteristics of Index Numbers
8.4 Uses of Index Numbers
8.5 Problems of Constructing Index Numbers
8.6 Common Notations
8.7 Methods of Constructing Index Numbers

8.7.1 Aggregative Method
8.7.2 Relative Method
8.8 Quantity Index Numbers
8.9 Tests of Adequacy of Index Number
8.9.1 Time Reversal Test
8.9.2 Factor Reversal Test
8.9.3 Circular Test
8.10 Errors in Index Numbers
8.10.1 Formula Error

8.2 Business Mathematics and Statistics

8.10.2 Homogeneity Error
8.10.3 Sampling Error
8.11 Chain Base Index Numbers
8.12 Base Shifting, Splicing and Deflating
8.13 Consumer Price Index or Cost of Living Index
8.13.1 Construction of Consumer Price Index Numbers
8.13.2 Methods of Construction
8.13.3 Applications and Uses
8.14 Illustrative Examples

8.1  INTRODUCTION

Economic activities have a tendency to change and it occurs due to the change in

an economic variable or a group of economic variables. It is necessary to measure

such change of economic activities with the help of a statistical device.

Index number is such a statistical device which measures percentage changes

of an economic variable or variables to derive a conclusion about the change of

the economic activities over a specified time. For example, we want to measure

the change in the average retail price of tea in 2017 with that in 2010 or we

want to measure the retail price of Darjeeling tea with that of Assam tea. We

may also like to know the increase in the yield of paddy in India during 2017 as

compared to 2010. We can easily measure such changes by expressing the price

or quantity of a given period as percentage of the price or quantity in the base

period (taken as 100), i.e. the period with which comparison is made. These

percentages which measure changes in the prices or quantities at a specified time

with that of the base period is known as index number. Suppose, average retail

price of tea per kg. in 2010 and 2017 are ` 120 and ` 150 respectively. Then index

number of tea for the year 2017 is 125 æ 150 ´ 100 ö , taking base year as 2010,
èç 120 ÷ø

means there is an increase of 25% in the average retail price of tea as compared

to the corresponding figure for the year 2010.

The problems arise when we want to measure changes in a complex phenomenon

like agricultural production or industrial production, wages, export, import, cost

of living, etc. This involves measuring changes in the prices or quantities of many

commodities. Since the prices or quantities of different commodities change by

different degrees or units, they can not be measured directly. For example, changes

in agricultural production in a country are not capable of direct measurement, but

it is possible to study relative changes in agricultural production by studying the

changes in the values of some such variables which affect agricultural production,

and which are capable of direct measurement. So, we have to consider all items of

Index Numbers 8.3

production for measuring the combined changes in a group related variables and
each item may have undergone a different fractional increase (or even a decrease).
Index number provides such composite measurement which may be defined as a
device for combining the changes or variations that have come in groups of related
variables over a period of time, with a view to obtain a figure that represents the
‘net’ result of the change in the constitute variables.

In reality, Index numbers are used to feel the pulse of the economy and they
reveal the inflationary or deflationary tendencies. Hence, index numbers are
described as barometers of economic activity. If one wants to have an idea as
to what is happening in an economy, he should check the important indicators,
like the index numbers of industrial production, agricultural production, business
activity, etc.

8.1.1 Meaning of Index Number
An index number is a statistical measure designed to show changes in a variables
or a group of related variables with respect to time, geographic location or other
characteristics such as income, profession, etc.

8.2  TYPES OF INDEX NUMBERS

There are three types of index numbers which are commonly used.

8.2.1 Price Index Numbers
Price index numbers measure the relative changes in prices of a commodity
between two periods. It is basically the ratio of the price of a certain number
of commodities in the present year as against base year. The Whole Price Index
(WPI), Consumer Price Index (CPI) are some of the popularly used price indices.

8.2.2 Quantity Index Numbers
Quantity Index Numbers measure the changes in the physical quantity of goods
produced, consumed or sold of an item or a group of items from one period to
another. The index of industrial production is the popularly used quantity index.

8.2.3 Value Index Numbers
The value index combines price and quantity changes to present a more spatial
comparison. It is also known as a combination index. These pertain to compare
changes in the monetary value of imports, exports, production or consumption of
commodities. It has limited use as it can not distinguish the effects of price and
quantity separately.

8.4 Business Mathematics and Statistics

8.3  CHARACTERISTICS OF INDEX NUMBERS

1. Index Numbers are Specialized Averages

An index number is an average with a difference. Averages can be used to compare
only those series which are expressed in the same units. Whereas index number is
obtained as a result of an average of all items which are expressed in different units.

2. Index Numbers Measure Changes that are not Directly
Measurable

An index number is used for measuring the magnitude of changes in phenomenon
that are not directly measurable. The cost of living, business activity in a country
are not directly measurable but it is possible to study relative changes in these
activities by measuring the changes in the values of variables/factors which
effect these activities.

3. Index Numbers are Expressed as Percentage

An index number is calculated as a ratio of the current value to a base value and
expressed in terms of percentages to show the extent of relative change. The
index number for the base year is always taken as 100.

4. Expressed in Number

Index number can be expressed in number only.

5. Universal Utility

Changes in the quantity of agricultural production, industrial production, imports
and exports can also measured through index numbers.

8.4  USES OF INDEX NUMBERS

• Helps in policy formulation: Index numbers measure trends of various
phenomena and on the basis of these trends and tendencies the government
can formulate different policies, such as determining the rates of dearness
allowance, price policies, etc.

• Helps in studying trends: Index numbers reveal a general trend of the pheno­
menon under study such as trend of exports, imports, national income, etc.

• Helps in determining purchasing power of money: Index numbers
measure the purchasing power of money and determine the real wages.

• Helps in deflating various values: Index numbers play a vital role in
adjusting the original data to reflect reality. For example, nominal income can
be transformed into real income by using income deflators. Index numbers
are also helpful in deflating national income on the basis of constant prices.

Index Numbers 8.5

• Helps in measuring effectiveness: Index numbers measure the effectiveness
of teaching system in the field of health. It can be used to show the general
health condition of the people and to indicate the adequacy of hospital facilities.

8.5  PROBLEMS OF CONSTRUCTING INDEX NUMBERS

Before constructing index numbers, a careful thought must be given to the
following problems:

• The purpose of the index: At the very outset the purpose of constructing
the index must be clearly decided. There cannot be any all­purpose index.
Every index is of limited and particular use. Thus, a price index that is
intended to measure consumers’ prices must not include wholesale prices.
Therefore, it is important to be clear about the purpose of the index number
before its construction.

• Selection of the items: The second problem in the construction of index
numbers is the selection of the items. The items which are related to and are
relevant with the purpose for which the index is constructed should be included.

• Selection of price quotation: After the items have been selected, the next
problem is to obtain price quotations for these items. It is a well­known fact
that prices of many items vary from place to place and even from shop to
shop in the same market. It is neither possible nor necessary to collect prices
of the items from all markets in the country where it is dealt with, we should
take a sample of the markets. A selection must be made to represent places
and persons. These places should be well known for trading these items.

• Selection of the base year: The base year is defined as that year with
reference to which the price changes in other years are compared and
expressed as percentages. It is therefore necessary that
(i) the base year should be normal, and
(ii) it should not be too far in the past.
There are two methods for selecting the base year. One method is the
selection of a certain year as a base year, known as fixed base method.
While the other is chain base method, where relatives of each year are
calculated on the basis of the prices of the preceding year. The chain base
index numbers are called Link Relatives.

• Selection of the average: As index numbers are themselves specialised
averages, it has to be decided first as to which average should be used for
their construction. The choice lies between arithmetic mean and geometric
mean alone as other measures of averages are rarely used. Theoretically,
geometric mean is the best for this purpose. But, in practice, arithmetic
mean is used because it is easier to follow.

• Selection of the weights: Generally, all the items included in the
construction of index numbers are not equally important. Therefore, if the
index numbers are to be representative, proper weights should be assigned
to various items in relation to their relative importance. A more important

8.6 Business Mathematics and Statistics

item will get more weight. Because of a rise in the prices of essential items
the poor consumer is more affected than others. Weights should be unbiased
and be rationally and not arbitrarily selected.
• Selection of an appropriate formula: A large number of formulae have
been devised for constructing the index. The problem is that of selecting
the most appropriate formula. The choice of the formula would depend on
the purpose of the index as well as on the data available.

8.6  COMMON NOTATIONS

The following notations will be used to derive the various mathematical formulae
of index number.

p = Price per unit in the base year (denoted by suffix o)

o

p = Price per unit in the current year (denoted by suffix n)

n

q = Quantity in the base year (denoted by suffix o)

o

q = Quantity in the current year (denoted by suffix n)

n

P = Price index number for the current or given year n with respect to the

on

base year o
Q = Quantity index number for the current or given year n with respect to

on

the base year o
I = Index number for the current or given year n with respect to the base year o

on

I = Index number for the current or given year o with respect to the base year n

no

8.7  METHODS OF CONSTRUCTING INDEX NUMBERS

Several methods have been considered for constructing index numbers. Principal
methods are illustrated in Chart 8.1.

Construction of Index Numbers

Aggregative method Relative method
Simple or unweighted
Aggregative method Simple Average
of Relatives
Weighted Aggregative method
Weight Average
Laspeyres’ Formula of Relatives
Paasche’s Formula
Edgeworth–Marshall’s Formula
Fisher’s Ideal Formula
Bowley’s Formula

Chart 8.1 Methods for Construction of Index Numbers

Index Numbers 8.7

In this section we shall be mainly interested in Price Index Numbers showing
changes with respect to time, although methods stated above can be applied to
other cases. In general, the present level of prices is compared with the level of
prices in the past. The present period is called the current period or year and
some period in the past is called the base period or year.

8.7.1 Aggregative Method

In aggregative method, the aggregate (average) price of all items in a given year

is expressed as a percentage of the aggregate (average) price of all items in the

base year, giving the index number. Hence,

Price Index Number = Aggregate price in the given or current year ×100
Aggregate price in the base year

(i) Simple or unweighted aggregative method: Under this method the total

of the current year prices for various items is divided by the total of the

base year and multiplying the result by 100.

Symbolically, S pn
S po
Ion = × 100 … (8.1)

ILLUSTRATION 1

Construct the price index number for 2017, taking the year 2010 as the base
year by using simple aggregative method:

Commodity Price in the year Price in the year

2010 2017

A 30 40

B 25 30

C 35 50

D 60 80

E 50 75

Solution: Calculation of price index number by simple aggregative method

Commodity Price in 2010 Price in 2017
A (`) (po) (`) (pn)
30 40

B 25 30

C 35 50

D 60 80

E 50 75

Total ∑po = 200 ∑pn = 275

8.8 Business Mathematics and Statistics

Price Index Number = S pn × 100 = 275 × 100 = 137.5
S po 200

Therefore, the price index for the year 2017, taking 2010 as base year, is
137.5, showing that there is an increase of 37.5% in the prices in 2017 as
against 2010.

(ii) Weighted aggregative method: This method is same as simple aggre­
gative method with the only difference that the weights are assigned to the
various items included in the index.
Symbolically,

Ion = S pnw × 100 [where w represents the ‘weight’] …(8.2)
S pow

The importance of the price of a commodity in the overall picture described

by an index number is generally determined by its quantity produced,

consumed, marketed or sold. We, therefore, use the quantities in the base

year or given year or the average of several years as weights. There are

various formulae of assigning weights to an index. The more important

ones are as follows:

(a) Laspeyres’ Formula: In this formula base year quantities (qo) are
used as weights. Therefore, substituting w by qo in equation 8.2 we
get Laspeyres’ price index formula.

Laspeyres’ Price Index (Ion) = S pnqo × 100 …(8.3)
S poqo

(b) Paasche’s Formula: In this formula current year quantities (qn) are
used as weights. Therefore, substituting w by qn in equation 8.2 we
get Paasche’s Price Index Formula.

Paasche’s Price Index (Ion) = S pnqn × 100 …(8.4)
S poqn

(c) Edgeworth–Marshall’s Formula: In this formula the average

of the quantities of the base year and the current year are used as

weights. Therefore, substituting w by æ qo + qn ö equation 8.2 we
çè 2 ø÷ in

get Edgeworth–Marshall’s Price Index Formula.

Index Numbers 8.9

Edgeworth–Marshall’s Price Index

S pn æ qo + qn ö
èç 2 ÷ø
(Ion) = + qn ö × 100 = S pnqo + S pnqn × 100 ...(8.5)
æ qo 2 ø÷ S poqo + S poqn
S po èç

(d) Fisher’s Ideal Formula: This formula is the geometric mean of
Laspeyres’ formula and Paasche’s formula and is given by:
Fisher’s Ideal Price Index

(Ion) = Laspeyres’ Price Index × Paasche’s Price Index

= S pnqo ´ S pnqn × 100 …(8.6)
S poqo S poqn

Fisher’s formula is known as ‘Ideal’ because of the following reasons:

(i) It is based on the geometric mean which is theoretically considered

the best average for the construction of index numbers.

(ii) Both the current year and base year prices and quantities are taken

into account by this index.

(iii) It satisfies both the time reversal test and factor reversal test, i.e.

the tests of adequacy of index numbers.

(iv) It neutralizes the upward bias of Laspeyres’ index and downward

bias of Paasche’s index to a great extent. In fact, Fisher’s ideal

index is free from any bias.

(e) Bowley’s Formula: This formula is the arithmetic mean of Laspeyres’

formula and Paasche’s formula and is given by:

Bowley’s Price Index (Ion) = 1
Price Index] [Laspeyres’ Price Index + Paasche’s
2

=1 é S pn qo + S pnqn ù × 100 …(8.7)
2 ê S po qo S poqn ú
ë û

ILLUSTRATION 2

Find the index number by using weighted aggregative method from the
following data:

Commodities Base Price (2014) Current Price (2017) Weight

Rice 36 54 10

Dal 30 50 3

Fish 130 155 2

Potato 40 35 4

Oil 110 110 5

8.10 Business Mathematics and Statistics

Solution: Calculation of Price Index Number by weighted aggregative method

Commodities Base Price Current Price Weight pow pnw
(2014) (po) (2017) (pn) (w)
Rice 360 540
Dal 36 54 10 90 150
Fish 30 50 3 260 310
Potato 130 155 2 160 140
Oil 40 35 4 550 550
Total 110 110 5 ∑pow = ∑pnw =
1420 1690

Price Index Number = S pnw × 100 = 1690 × 100 = 119.01
S pow 1420

Therefore, the price index for the year 2017, taking 2014 as base year is 119.01.

ILLUSTRATION 3

Using (i) Laspeyres’ (ii) Paasche’s (iii) Edgeworth­Marshall (iv) Fisher
(v) Bowley’s formulae, find the index number from the following data:

Commodity Price per unit Unit

A Base Year Current Year Base Year Current Year
B
C 4 10 50 40

3 8 10 8

2 4 54

Solution: Calculation of Price Index Numbers

Commodity Base Current Base Current
year year year year qty. poqo pnqo poqn pnqn
A price price qty.
B (po) (pn) (qo) (qn)
C 10 50
Total 4 40 200 500 160 400
8 10 8 30 80 24 64
3 4 10 20 8 16
4 5 – 240 600 192 480
2
– –
–

Index Numbers 8.11

(i) Laspeyres’ Price Index Number = S pnqo × 100 = 600 × 100 = 250
S poqo 240

(ii) Paasche’s Price Index Number = S pnqn × 100 = 480 × 100 = 250
S poqn 192

(iii) Edgeworth-Marshall’s Price Index Number

= S pnqo + S pnqn × 100 = 600 + 480 × 100 = 1080 ×100 = 250
S poqo + S poqn 240 + 192 432

(iv) Fisher’s Price Index Number = Laspeyres¢ Index ´ Paasche¢s Index

= 250 ´ 250 = 250

(v) Bowley’s Price Index Number = 1 [Laspeyres’ Index + Paasche’s
2 Index]
= 1 [250 + 250] = 250
2

8.7.2 Relative Method

Price Relative

The price of each item in the current year is expressed as a percentage of price
in base year. This is called price relative and expressed as the following formula:

Price Relative = Price in the given or current year × 100 = Pn × 100
Price in the base year po

(i) Simple Average of Relatives Method: In this method, the price relatives

for all items is calculated and then averaged to get the index number. Thus,

(a) If Arithmetic Mean (A.M.) is taken as average, then we have simple

S æ pn ´ 100 ö
ç ÷
è po ø [where N is the
A.M. of Relative Index Number (Ion) = N
number of items]

(b) If Geometric Mean (G.M.) is taken as average, then we have Simple

G.M. of Relative Index Number (Ion) = N (Product of Price Relatives)

(ii) Weighted Average of Relatives Method In this method, either current
year values (pn.qn) or base year values (poqo) are used as weight. The index

8.12 Business Mathematics and Statistics

number for the current year is calculated by dividing the sum of the products
of the current year’s price relatives and base year or current year values by
the total of the weights. Thus,

(a) If Arithmetic Mean (A.M.) is taken as average, then we have

weighted A.M. of Relative Index Number (Ion) =
æ ö
S ç pn ´ 100 ÷.V

è po ø or SP.V
SV
SV
where V = value weights and P = pn × 100
po

Taking V = poqo, Laspeyres’ Index Number and V = poqn, Paasche’s

Index Number can be derived from above formula.

(b) If Geometric Mean (G.M.) is taken as average, then we have weighted

G.M. of Relative Index Number (Ion) = antilog é S (log P).V ù
êë SV ûú

where P = pn × 100 for every item, and V = value weight.
po

ILLUSTRATION 4

Compute price index using simple average and weighted average of price
relatives by applying (a) A.M. and (b) G.M.

Commodity Base year price Current year price Base year quantity
(`) (`)
P 2
Q 52 71 4
R 12 16 3
S 30 27 1
5 6

Solution: Calculation of Price Index Numbers

Commodity Base Current Base V= P = pn P.V. log P V.log P
year year year poqo po
price price qty.
× 100
(po) (pn) (qo)

P 52 71 2 104 136.54 14200.16 2.1352 222.0608

Q 12 16 4 48 133.33 6399.84 2.1250 102.0000

R 30 27 3 90 90.00 8100.00 1.9542 175.8780

S 5 6 1 5 120.00 600.00 2.0792 10.3960

Total – – – 247 479.87 29300.00 8.2936 510.3348

Index Numbers 8.13

(i) Simple A.M. of Price Relative Index Number

S æ pn ´ 100 ö
ç ÷
= è po ø = 479.87 = 119.9675
N4

(ii) Simple G.M. of Price Relative Index Number

= n Product of Price Relatives

= 4 136.54 ´ 133.33 ´ 90 ´ 120

Let I be the Index Number

Therefore, I = 4 136.54 ´ 133.33 ´ 90 ´ 120

1

= (136.54 ´ 133.33 ´ 90 ´ 120)4

or log I = 1 (log 136.54 + log 133.33 + log 90 + log 120)
4

= 1 ´ S log P
4

= 1 × 8.2936
4

= 2.0734

Therefore, I = antilog 2.0734

= 118.4

Therefore, Price Index Number = 118.4

(iii) Weighted A.M. of Price Relative Index Number

S æ pn ´ 100 ö
ç ÷V
è po ø S PV 29300
= = SV = 247 = 118.62
SV

(iv) Weighted G.M. of Price Relative Index Number

= antilog é S (log P).V ù = antilog æ 510.3348 ö = antilog (2.066)
ëê SV úû çè 247 ÷ø

= 116.4.

8.14 Business Mathematics and Statistics

8.8  QUANTITY INDEX NUMBERS

Price index numbers measure the changes in the price of certain goods. Quantity
index numbers, on the other hand, measure the changes in the volume of produc­
tions, construction or employment over a period of years. Methods of construction
of quantity index number are similar to those involved in price index numbers.
Quantity index numbers can be obtained easily by changing p to q and q to p in
the various formulae of price index numbers.

Method Price Index Quantity Index
I. Aggregative Number (Pon) Number (Qon)
A. Simple aggregative
S pn ´ 100 S qn ´ 100
S po S qo

B. Weighted aggregative S pnw ´ 100 S qnw ´ 100
S pow S qow

(i) Laspeyres’ Formula S pnqo ´100 S qn po ´100
(ii) Paasche’s Formula S poqo S qo po

S pnqn ´ 100 S qn pn ´ 100
S poqn S qo pn

(iii) Fisher’s Formula S pnqo ´ S pnqn ´ 100 S qn po ´ S qn pn ´ 100
S poqo S poqn S qo po S qo pn
(iv) Edgeworth­
Marshall’s Formula S pn (qo + qn ) ´ 100 S qn ( po + pn ) ´ 100
S po (qo + qn ) S qo ( po + pn )

1 éS pnqo + S pnqn sù ´ 1 é S qn po + S qn pn ùú´100
(v) Bowely’s Formula 2 ê poqo S poqn ú 100 2 ê S qo po S qo pn û
II Relative ë S û ë

A. Simple A.M. of S æ pn ´100 ö S æ qn ´ 100 ö
Relative Index ç ÷ ç ÷
è po ø è qo ø

N N

B. Weighted A.M. of S æ pn ´ ö w S æ qn ´ ö w
Relative Index çè po 100ø÷ èç qo 100ø÷

Sw Sw

Index Numbers 8.15

ILLUSTRATION 5

Calculate Quantity Index Numbers from the following data using
(a) Laspeyres’ Formula (b) Paasche’s Formula and (c) Fisher’s Formula.

Items Base year Base year Current year Current year
Price (`) Quantity (kg.) Price (`) Quantity (kg.)
(pn)
(po) (qo) (qn)

P6 52 11 58

Q 4 102 5 122

R 5 62 7 62

S 12 32 15 26

T8 42 11 38

Solution: Calculation of Quantity Index Numbers

Items po qo pn qn poqo pnqn pnqo poqn

P 6 52 11 58 312 638 572 348
Q 4 488
R 5 102 5 122 408 610 510 310
S 12 312
T 8 62 7 62 310 434 434 304
Total – 1762
32 15 26 384 390 480

42 11 38 336 418 462

– – – 1750 2490 2458

(a) Laspeyres’ Quantity Index Number

(Qon) = S qn po × 100 = 1762 × 100 = 100.69
S qo po 1750

(b) Paasche’s Quantity Index Number

(Qon) = S qn pn × 100 = 2490 × 100 = 101.3
S qo pn 2458

(c) Fisher’s Quantity Index Number

(Qon) = S qn po ´ S qn pn × 100
S qo po S qo pn

= 1762 ´ 2490 × 100 = 100.99 = 101 (Approx.)
1750 2458

8.16 Business Mathematics and Statistics

8.9  TESTS OF ADEQUACY OF INDEX NUMBER

We have discussed several methods for constructing simple and weighted index
numbers. The question arises which method of index number is the most suitable
in a given situation. However, some tests have been suggested to determine the
adequacy of a method of index number. These tests are:

8.9.1 Time Reversal Test

According to Fisher the formula for calculating the index number should be such
that it will give the same ratio between one point of comparison and the other,
no matter which of the two is taken as base. This means that the index number
should work both backwards as well as forwards. When the data for any two
years are treated by the same method, but with the base reversed, the two index
numbers should be reciprocals of each other and hence, their product should be
unity. Mathematically, the index number (Ion) for the year n with respect to the
base year o is reciprocal of the index number (Ino) for the year o with respect to
the base year n. Symbolically,

Ion = 1 or Ion × Ino = 1 (omitting the factor 100 from each index)
Ino

The index number formula which obey the above relation is said to satisfy the
Time Reversal Test. This test is satisfied by

(i) Simple aggregative formula
(ii) Simple geometric mean of price relatives formula
(iii) Weighted geometric mean of price relatives formula
(iv) Edgeworth–Marshall’s formula
(v) Fisher’s ideal index formula

8.9.2 Factor Reversal Test

Another test proposed by Fisher is known as Factor Reversal Test. In the
words of Fisher, just as each formula should permit the interchange of the
two times without giving inconsistent results, similarly it should permit inter­
changing the prices and quantities without giving inconsistent results which
means two results multiplied together should give the true value ratio. The test
says that the change in price multiplied by change in quantity should be equal
to total change in value. In other words, an index number formula is said to
satisfy factor reversal test if the product of the price index (Pon) and quantity
index (Qon) equals the total value ratio of the current year and the base year.
Symbolically,

Index Numbers 8.17

Pon × Qon = S pnqn where S pnqn = value ratio
S poqo S poqo

This test is satisfied only by Fisher’s ideal index number formula.

8.9.3 Circular Test

Another test of adequacy applied in index number studies is the circular test. This
test was suggested by Westerguard and C.M. Walsch. It is an extension of time
reversal test for more than two years and based on the shift ability of the base.
Accordingly, the index should work in a circular fashion. Thus, if Io1 be the index
number for the year 1 with respect to the base year o, I12 be the index number for
the year 2 with respect to the base year 1 and so on, In-1, n, the index number for
the year n with respect to the base year n – 1 and Ino the index number for the year
o with respect to the base year n then the circular test is satisfied if

I01 × I12 × I23 × ……………….× I(n–1), n × Ino = 1

This test is satisfied by

(i) Simple aggregative index number formula
(ii) Simple G.M. of relatives formula

ILLUSTRATION 6

Show that neither Laspeyres’ formula nor Paasche’s formula obeys Time
Reversal and Factor Reversal Tests of Index Numbers.

[C.U. B.Com. 2012, 2014, 2015]

Proof:
1. Time Reversal Test:

(a) Laspeyres’ index number formula (Ion) = S pnqo (omitting the factor
100) S poqo

Interchanging ‘o’ and ‘n’ we have,

Ino = S poqn
S pnqn

Now, Ion × Ino = S pnqo × S poqn ≠1
S poqo S pnqn

thus, Laspeyres’ formula does not obey time reversal test.

8.18 Business Mathematics and Statistics

(b) Paasche’s index number formula (Ion) = S pnqn (omitting the factor
100) S poqn

Interchanging ‘o’ and ‘n’ we have,

Ino = S poqo
S pnqo

Then, Ion × Ino = S pnqn × S poqo ≠1
S poqn S pnqo

thus, Paasche’s formula also does not obey time reversal test.

2. Factor Reversal Test:
(a) Laspeyres’ price index number formula

(Pon) = S pnqo (omitting the factor 100)
S poqo

Interchanging ‘p’ and ‘q’, we get Laspeyres’ Quantity index formula

(Qon) = S qn po = S poqn
S qo po S poqo

Then, Pon × Qon = S pnqo × S poqn ≠ S pnqn (value ratio)
S poqo S poqo S poqo

Thus, Laspeyres’ formula does not obey factor reversal test.

(b) Paasche’s Price Index formula (Pon) = S pnqn (omitting the factor
100) S poqn

Interchanging ‘p’ and ‘q’, we get,

Paasche’s Quantity Index formula (Qon) = S qn pn = S pnqn
S qo pn S pnqo

Then, Pon × Qon = S pnqn × S pnqn ≠ S pnqn (value ratio)
S poqn S pnqo S poqo

Thus, Paasche’s formula also does not obey factor reversal test.

ILLUSTRATION 7

Prove that Fisher’s Ideal Index number satisfies both time reversal test and

factor reversal test. [C.U. B.Com. 2009, 2013 (H), 2014 (G)]

Index Numbers 8.19

Proof:
1. Time Reversal Test:
Fisher’s Ideal Index Number Formula

(Ion) = S pnqo ´ S pnqn (omitting the factor 100)
S poqo S poqn

Interchanging ‘o’ and ‘n’, we have

Ino = S poqn ´ S poqo
S pnqn S pnqo

Then, Ion × Ino = S pnqo ´ S pnqn ´× S poqn ´ S poqo
S poqo S poqn S pnqn S pnqo

= S pnqo ´ S pnqn ´ S poqn × S poqo
S poqo S poqn S pnqn S pnqo

= 1 =1
Thus, Fisher’s Ideal Index Number satisfies time reversal test.

2. Factor Reversal Test:
Fisher’s Price Index Number formula

Pon = S pnqo ´ S pnqn (omitting the factor 100)
S poqo S poqn

Interchanging ‘p’ and ‘q’, we have

Fisher’s Quantity Index Number formula

Qon = S qn po ´ S qn pn = S poqn ´ S pnqn
S qo po S qo pn S poqo S pnqo

Then, Pon × Qon = S pnqo ´ S pnqn ´ S poqn ´ S pnqn
S poqo S poqn S poqo S pnqo

= S pnqn ´ S pnqn = S pnqn
S poqo ´ S poqo S poqo

Thus, Fisher’s Ideal Index Number satisfies factor reversal test.

ILLUSTRATION 8
Prove that the simple aggregative formula satisfies Circular Test.

8.20 Business Mathematics and Statistics

Proof: We know,

Simple aggregative index number for the year 1 with base year 0

Sp
I01 = 1 (omitting the factor 100)

S p0

Simple aggregative index number for the year 2 with base year 1

I12 = S p2 (omitting the factor 100)
S p1

Simple aggregative index number for the year 3 with base year 2

I23 = S p3 (omitting the factor 100)
S p2
and so on,

Simple aggregative index number for the year n with base year (n – 1)

I(n – 1), n = S pn (omitting the factor 100)
S pn-1
and,

Simple aggregative index number for the year 0 with base year n

Ino = S po (omitting the factor 100)
S pn

Clearly, I01 × I12 × I23 × …………………….. × I(n – 1), n × Ino

= S p1 ´ S p2 ´ S p3 ´ ............. ´ S pn ´ S po =1
S p0 S p1 S p2 S pn-1 S pn

Thus, simple aggregative formula satisfies circular test.

8.10  ERRORS IN INDEX NUMBERS

Errors may crop up in the construction of index numbers due to faulty selection
of items, inadequate information about price quotation and due to lack of repre­
sentative character of price quotation, etc. Index numbers are mainly affected by
the following three types of errors:

8.10.1 Formula Error

Different formulae are used for the calculation of index numbers and as such we
get different values of an index number. There is no specific formula which can
be considered the most suitable formula for the construction of an index number.
Each formula used introduces an error which can never be eliminated. This error
is known as formula error.

8.10.2 Homogeneity Error

For the construction of index numbers same commodities are considered at
the base year and at the current year. But in reality, due to the change of taste,

Index Numbers 8.21

consumption habit, obsolescence, commodities consumed during the long period
between base year and current year may not be the same. Hence, the homoge­
neity in the composition of the commodities can not be strictly maintained and
as such are not comparable. This introduces an error called homogeneity error.

8.10.3 Sampling Error

All items not considered for the construction of index number. It is done, only on the
basis of sample items. As many items are left out during the construction process, the
calculated index number do not represent the actual changes in the phenomenon. The
error thus occurred due to the consideration of sample items is known as sampling error.

8.11  CHAIN BASE INDEX NUMBERS

The various formulae discussed so far assume that the base period remains the
same throughout the series of the index. This method though convenient has certain
limitations. The index of a given year on a given fixed base year is not affected by
changes in the prices or the quantities in all the intermediate years between two
periods or years. New items may have to be included and old ones may have to
be deleted in order to make the index more representative. In such cases it may be
desirable to use the chain base method. To construct index numbers by chain base
method, a series of index numbers are computed for each year with preceding year
as the base. These index numbers are known as link relatives or link indices.

Therefore, link index for nth year = Index number of nth year with (n – 1)th
year as base.

Let po, p1, p2, p3,… represent prices during the successive years, denoted by
0,1,2,3,… Then, the link indices of each year will be as under

I¢01 = p1 × 100, I¢12 = p2 × 100, I¢23 = p3 × 100, …
p0 p1 p2

These link indices or link relatives when multiplied successively known as

the chaining process give link to a common base. The products thus obtained are

expressed as percentage and give the required index number. This index number

is called chain base index number. For example, the chain base index number

for the nth year with 0th year as base is given by (expressed as percentages)

Ion = I¢01 × I¢12 × I¢23 × ... × I¢(n-2)(n-1) ´ I¢(n-1), n

= I¢0, (n -1) ´ I¢(n-1), n

= [chain base index number for (n – 1)th year with 0th year as
base] × [link index of the nth year with (n – 1)th year as base]

(Here Iʹ is used for link index and I is used for chain base index)
thus,

8.22 Business Mathematics and Statistics

Chain base index for any year

= (chain base index of previous year) × (link index of the current year)
100

i.e.
I01 = I¢01, I02 = I01 ´ I¢12
I03 = I02 × I′23 and so on.

Therefore, the steps of chain base index are:
(i) Express the figures of each year as a percentage of the preceding year to

obtain link relatives or link indices.
(ii) The link relatives are chained together by successive multiplication to get a

chain index.
Advantages:

(1) In this method the price relatives of a year can be compared with the price
levels of the immediately preceding year. Businesses mostly interested
in comparing this time period rather than comparing rates related to the
distant past will utilize this method.

(2) In this method it is possible to include new items in an index number or to
delete old items which are no longer important which is not possible with
the fixed base method.

(3) The effects of all intermediate years between two periods are taken into
consideration.

(4) The weights of the different items can be adjusted frequently.
(5) Index numbers computed by this method are free to a great extent from

seasonal variation than those obtained by the other methods.
Disadvantages:

(1) Under this method comparisons cannot be made over a long period because
the long range comparisons of chained percentages are not strictly valid.

(2) This method requires tedious calculations and hence time consuming.
(3) The significance of index numbers obtained by this method is difficult to

understand.

ILLUSTRATION 9

Construct Index Numbers by chain base method from the following data of
wholesale prices:

Year: 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014
Prices: 75 50 65 60 72 70 69 75 84 80

Index Numbers 8.23

Solution: Computation of chain base index numbers
Year
2005 Prices Link relatives Chain base index numbers
2006
2007 75 100 100
2008 50 50 ´100 = 66.67 66.67 ´100
2009
2010 75 = 66.67
2011 100
2012
2013 65 65 ´100 = 130 130 ´ 66.67
2014 50 = 86.67

100

60 60 ´100 = 92.31 92.31 ´ 86.67
65 100 = 80

72 72 ´100 = 120 120 ´ 80 = 96
60 100

70 70 ´100 = 97.22 92.22 ´ 96
72 = 93.33

100

69 69 ´100 = 98.57 98.57 ´ 93.33
70 = 92

100

75 75 ´100 = 108.69 108.69 ´ 92
69 = 100

100

84 84 ´100 = 112 112 ´100
75 = 112

100

80 80 ´100 = 95.24 95.24 ´112 = 106.67
84 100

8.12  BASE SHIFTING, SPLICING AND DEFLATING

Base Shifting: Sometimes, it becomes necessary to change the base year of a
series of index numbers from one period to another for the comparison. This
change of reference base period is usually referred to as
“Shifting the base”. Shifting of base is generally required due to the following
reasons:

(i) The base year is too old to compare the current year and we want to choose
a recent year as new base to make the index numbers more useful.

8.24 Business Mathematics and Statistics

(ii) If different series of index numbers are based on different base years and
they are to be compared from each other. In this case all the series are
expressed with respect to a common base.

Under these circumstances it is necessary to recompute all index numbers using
new base year. Such computation of index numbers using new base year is to
divide index number in each year by the index number corresponding to the new
base year and then to express the result as percentages.

Symbolically,
Index number (with new base year)

Old index number for the current year
= ×100

Old index number for the new base year

[ 100 is known as multiplying factor]
Old index number for the new base year

ILLUSTRATION 10

The following table represents the price index number of a commodity with
base 2008. Shift the base to 2015

Year: 2012 2013 2014 2015 2016

Index number: 120 145 165 175 190

Solution: Shifting of base from 2008 to 2015

Year Index Number Index Number
(Base – 2008) (Base – 2015)

2012 120 120 ´100 = 68.57
175

2013 145 145 ´100 = 82.86
175

2014 165 165 ´100 = 94.28
175

2015 175 175 ´100 = 100
175

2016 190 190 ´100 = 108.57
175

[In this calculation 100 is the multiplying factor.]
175

Index Numbers 8.25

Splicing: Sometimes, a series of index numbers with an old base is discontinued
and a new series of index numbers is constructed by taking some recent year
as base. In this situation index numbers of these two series are not comparable
because both are based on different years. In order to make them comparable it
is necessary to convert these series of index numbers of different bases into a
continuous series of index numbers of a common base.

The statistical procedure which convert an old index number with a new one
is called Splicing. Splicing arises only, if the following conditions are satisfied:

(i) If the old and the new series of index numbers have been constructed with
the same items;

(ii) If the old and the new series of index numbers have different base year; and
(iii) If the old and the new series of index numbers have at least one overlapping

year.
Types of Splicing: Splicing are of two types:

(i) Forward Splicing: When the new indices are to be spliced with the base
year of the old indices, then it is called forward splicing. The following
formula is used for computation:
Spliced Index Number (from new to old)

New Index Number ´ Old Index Number
= for the current year for the new base year

100

é Old Index Number for the new base year is known as multiplying ù
êë 100 factorúû

(ii) Backward splicing: When the old indices are to be spliced with the base

year of the new indices, then it is called backward splicing. The following

formula is used for computation:

Spliced index number (from old to new)

Old Index Number for the current year
= ×100
Old Index Number for the new base year

é 100 ù
êë Old Index Number for the new base year is known as multiplying factor.ûú

ILLUSTRATION 11

Two series of price index numbers are given below. Splice them on the new
base 2012 and also on the old base 2006.

Year 2008 2009 2010 2011 2012 2013 2014 2015 2016
130 145 161 169 185 125 140 155 175
Series A
(Base – 2006) 100

Series B
(Base – 2012)

8.26 Business Mathematics and Statistics

Solution: Splicing of two series of Index Numbers

Year Series A Index Series B Index Spliced Index Numbers
Base Year -2006 Base Year -2012
Numbers Numbers

(Base – 2006) (Base – 2012)

2008 130 130 ´100
130 = 70.27

185

2009 145 145 ´ 100
145 = 78.38

185

2010 161 161 161´100
2011 169 = 87.03
185

169 ´100
169 = 91.35

185

2012 185 100 185 100
2013 125
2014 125 125 ´ 185 140
2015 = 231.25 155
2016 100 175

140 140 ´185
= 259.00
100

155 155 ´ 185
= 286.75
100

175 175 ´ 185
= 323.75
100

Deflating: Deflating means making allowances for the changes in the purchasing

power of money due to change in general price level. A rise in price level means

a reduction in the purchasing power of money. The process of adjusting a series

of wages or income according to current price changes to find out the level of

real wages or income is called deflating. The following formula is used to find

out real wages or income.

Real wages or income = Money wages or income ×100

Price Index or Consumer Price Index

The adjusted figures of wages or income can be converted into index numbers

for the purposes of comparison. It is known as Real wage (or income) indices.

Index Numbers 8.27

Such indices show changes in the purchasing power of the money wages (or
income) of the workers. Technically, it is known as deflating the index number.

Real wage (or income) Index Number or deflated Index Number
= Real wage (or income) of current year × 100
Real wage (or income) of base year

= Index of Money wage (or income) ×100
Price Index or Consumer Price Index

ILLUSTRATION 12

The following table gives the annual income of a worker and the general price
index numbers during 2008–2016. Calculate (i) Real Income and (ii) Real
Income Index Number with 2008 as base.

Year 2008 2009 2010 2011 2012 2013 2014 2015 2016
7200 8400 10,000 11,000 12,000 12,800 13600 14400 15000
Income
(`) 100 120 150 160 250 320 450 530 600

General
Price
Index
Number

Solution: Calculations of (i) Real Income and (ii) Real Income Index Number

Year Income General Real Income Real Income Index
(`) Price Index
(x) Number (I) æ y= x ö Number = æ y ´ 100 ö
èç I × 100 ÷ø èç 7200 ÷ø

(Base 2008 = 100)

2008 7200 100 7200 ´100 = 7200.00 100
100

2009 8400 120 8400 ´100 = 7000.00 97.22
120

2010 10,000 150 10,000 ´100 = 6666.67 92.59
150

2011 11,000 160 11,000 ´100 = 6875.00 95.49
160

8.28 Business Mathematics and Statistics

2012 12,000 250 12000 ´100 = 4800.00 66.67
2013 12,800 320 250 55.55
2014 13,600 450 41.97
2015 14,400 530 12800 ´100 = 4000.00 37.73
2016 15,000 600 320 34.72

13600 ´100 = 3022.22
450

14400 ´100 = 2716.98
530

15000 ´100 = 2500.00
600

8.13  CONSUMER PRICE INDEX OR COST OF LIVING INDEX

Different classes of people in a society consume different types of commodities
and if the same type of commodities then in different proportions. A change in
the level of prices affect different classes of people in different manners. General
index numbers fail to give an exact idea of the effect of the change in the general
price level on the cost of living of different classes of people. This necessitates
the construction of special purpose index numbers known as Consumer Price
Index Numbers. These indices are designed to measure the effect of changes
in the price of group of commodities and services on the purchasing power of
a particular class of society during any given period with reference to some
fixed base. The group of commodities and services will contain items like Food,
House Rent, Education, Clothing, Fuel and Light, Miscellaneous like Transport,
Washing, Newspaper, etc. Consumer Price Index Numbers are also called cost of
living index numbers or Retail price index numbers.

8.13.1 Construction of Consumer Price Index Numbers

The following steps are involved in the construction of consumer price index
numbers.

(1) Class of people: The first step in the construction of consumer price index
is that the class of people should be defined clearly. As far as possible a
homogeneous group of persons (e.g. industrial workers, officers, school
teachers, etc. residing in a particular well­defined area) regarding their
income and consumption pattern are considered. It is therefore necessary
to specify the class of people and locality where they reside.

Index Numbers 8.29

(2) Family budget inquiry and allocation of weights: The next step is to
conduct a family budget inquiry of the category of people concerned.
For this purpose some families are selected at random and inquiries are
conducted to determine the goods and services to be included in the
construction of index numbers. The inquiry includes questions on family
size, income, the quality and quantity of resources consumed and money
spent on them under various headings, such as clothing and footwear, fuel
and lighting, housing, etc. and the weights are assigned in proportions to
the expenditure on different items. This step has many practiced problems
as no two families have the same income and consumption pattern.

(3) Collection of consumer prices: The next step is to collect data on the
retail prices of the selected commodities for the current period and the base
period from the locality where the people reside or from where they make
their purchase. Collection of retail prices is not a easy task as the prices
vary from place to place and from shop to shop.
Finally, consumer price index numbers are computed by using suitable
index number formulae.

8.13.2 Methods of Construction
Following two methods are used to construct consumer price index numbers:

8.13.2.1 Aggregative Expenditure Method

In this method, the prices of commodities of current year as well as base year

are multiplied by the quantities consumed in the base year. The aggregate expen­

diture of current year is divided by the aggregate expenditure of the base year and

the quotient is multiplied by 100. Symbolically,

Consumer Price Index (Pon) = Spn qo × 100
where, Spo qo

S pnqo = Aggregative expenditure of the current year

S poqo = Aggregative expenditure of the base year.

This method is same as Laspeyres’ method.

8.13.2.2 Family Budget Method

In this method, the price relatives for each commodity are obtained and these

price relatives are multiplied by the value weights for each item and the product

is divided by the total of weights. Symbolically,

S æ pn ö poqo
ç po ÷
è ø
Consumer Price Index = ´ 100
S poqo

8.30 Business Mathematics and Statistics

= S I ´W
SW

where, Price Relative (I) = pn × 100, W = poqo
po

8.13.3 Applications and Uses

(i) Consumer price index numbers are primarily used for the calculation of

dearness allowance (D.A.) to maintain the same standard of living as in the

base year.

(ii) These index numbers are used for formulation of wage policy, general

economic policy, taxation at government level.

(iii) Purchasing power of money can be measured with the help of consumer

price index numbers. 1

Purchasing power of money = Consumer Price Index Number

(iv) Real wage (or income) can be measured by dividing the Actual wage (or

income) received during a period by the corresponding consumer price

index number of that period.

Real wage (or income) = Actual wage (or income) ´ 100
Consumer Price Index Number

ILLUSTRATION 13

Compute the cost of living index number using both the Aggregate Expen­
diture Method and Family Budget Method, from the following information:

Commodity Unit consumption in Price in Price in
base year
base year current year

Wheat 200 1.00 1.20
Rice 50 3.00 3.50
Pulses 50 4.00 5.00
Ghee 20 20.00 30.00
Sugar 40 2.50 5.00
Oil 50 10.00 15.00
Fuel 60 2.00 2.50
Clothing 40 15.00 18.00

[Calicut University, B.Com. 1987]

Index Numbers 8.31

Solution: Calculation of cost of living index

Commodity qo po pn pnqo w = pnqo I = pn ´ 100 I.W
po

Wheat 200 1.00 1.20 240 200 120 24,000.00

Rice 50 3.00 3.50 175 150 116.67 17,500.50

Pulses 50 4.00 5.00 250 200 125 25,000.00

Ghee 20 20.00 30.00 600 400 150 60,000.00

Sugar 40 2.50 5.00 200 100 200 20,000.00

Oil 50 10.00 15.00 750 500 150 75,000.00

Fuel 60 2.00 2.50 150 120 125 15,000.00

Clothing 40 15.00 18.00 720 600 120 72,000.00

Total 3085 2270 3,08,500.50

Cost of Living Index Number (CLI):

(a) Under Aggregate Expenditure Method

CLI = S pnqo ´ 100 = 3085 ´ 100 = 135.9
S poqo 2270

(b) Under Family Budget Method

CLI = S IW = 308500.50 = 135.9
SW 2270

ILLUSTRATIVE EXAMPLES
GROUP A: SHORT ESSAY TYPE

EXAMPLE 1

Prepare simple aggregative price index number from the following data:

Commodity Rate unit Price (2008) (`) Price (2017) (`)
Wheat per 10 kg 200 280
Rice per 10 kg 400 500
Pulses per 10 kg 500 700
Sugar 28 40
Oil per kg 80 100
per litre

8.32 Business Mathematics and Statistics

Solution: Calculation of simple aggregative price index number

Commodity Rate unit Price (2008) (`) Price (2017) (`)
Wheat per 10 kg 200 280
Rice per 10 kg 400 500
Pulses per 10 kg 500 700
Sugar 28 40
Oil per kg 80 100
Total per litre 1208 1620

Simple Aggregative Price Index Number
S
= S pn ´ 100 = 1620 × 100 = 134.1
po 1208

EXAMPLE 2

Find the price index number by using weighted aggregative method from the
following data:

Price per unit (`)

Commodity Base year Current year Weight

A (2015) (2017) 80
B 50
C 32 40 10
D 40
E 80 120 20

22

10 11

43

Solution: Calculation of weighted aggregative price index number

Commodity Base year Current year Weight po.w po.w
price (po) price (pn) (w)
A 40 2560 3200
B 32 120 80 4000 6000
80 50

C 2 2 10 20 20
D 10 11 40 400 440
E 4 3 20 80 60
Total
7060 9720

Weighted Aggregative Price Index Number
S pnw
= S pow × 100 = 9720 ´ 100 = 137.68
7060

Index Numbers 8.33

EXAMPLE 3

Compute the index number for the years 2011, 2012, 2013 and 2014, taking 2010
as base year, from the following data:

Year 2010 2011 2012 2013 2014
Price 120 144 168 204 216

Solution: Calculation of Price relative index numbers for different years

Year Price relative index number

2010 120 ´100 = 100
120

2011 144 ´100 = 120
120

2012 168 ´100 = 140
2013 120
2014 204 ´100 = 170
120
216 ´100 = 180
120

EXAMPLE 4

Find price index number by the method of relatives using arithmetic mean from
the following data:

Commodities: Wheat Milk Fish Sugar

Base Price : 5 8 25 6

Current Price: 7 10 32 12

[C.U. B.Com. 2015 (H)]

Solution: Calculation of Price Index Number

Commodities Base Price Current Price Pn ´ 100
(po) (pn) Po
Wheat
Milk 5 7 140
Fish 8 10 125
Sugar 25 32 128
Total 6 12 200
593

8.34 Business Mathematics and Statistics

Simple Arithmetic Mean of Price Relative

S æ pn ´ 100 ö
ç ÷
Index Number = è po ø = 593 (n = number of commodities = 4)
n4

= 148.25

EXAMPLE 5

Using the data given below calculate the Laspeyres’ Price Index Number for the
year 2014 with the year 2011 as base year:

Commodities Price (`) Quantity (kg)

2011 2014 2011 2014

A 4 5 95 120
B 60 70 118 130
C 35 40 50 70

Solution: [C.U. B.Com. 2015 (G)]
Calculation of Price Index Number

Commodities Price (`) Quantity (kg) pnqo poqo
2011 (po) 2014 (pn) 2011 (qo) 2014 (qn)
A 475 380
B 45 95 120 8260 7080
C 60 70 118 130 2000 1750
Total 35 40 50 70 10735 9210

Laspeyres’ Price Index Number = S pnqo ´ 100
S poqo

= 10735 ´ 100
9210

= 116.56

EXAMPLE 6

Using the data given below calculate the Price Index Number for the year 2013
by Paasche’s formula with the year 2010 as base:

Commodity Price per unit (`) Quantity (’000 kg)

2010 2013 2010 2013

Rice 9.3 14.5 100 90

Wheat 6.4 13.7 11 19
Pulse 3
5.1 12.7 5

[C.U. B.Com. 2014 (H)]

Index Numbers 8.35

Solution: Calculation of Price Index Number

Commodity Price per unit (`) Quantity (’000 kg) pnqn poqn
2010 (po) 2013 (pn) 2010 (qo) 2013 (qn)
Rice 1305 837
Wheat 9.3 14.5 100 90 260.3 121.6
Pulse 6.4 13.7 11 19 38.1 15.3
Total 5.1 12.7 53 1603.4 973.9

Paasche’s Price Index Number = S pnqn ´ 100
S poqn

= 1603.4 ´ 100
973.9

= 164.64

EXAMPLE 7

Calculate the Fisher’s Ideal Index Number from the following:

Commodities 2005 2010
Quantity (kg) Price (`)
Quantity (kg) Price (`)

Wheat 10 100 6 110
Rice 15 150 18 170
Cloth 50 5 30 4

Solution: [C.U. B.Com. 2015 (G)]

Calculation of Price Index Number

2005 2010

Commodities Quantity Price Quantity Price pnqo poqo pnqn poqn
(kg) (`) (kg) (`)
Wheat 1000 660 600
Rice (qo) (po) (qn) (pn) 2250 3060 2700
Cloth 250 120 150
Total 10 100 6 110 1100 3500 3840 3450

15 150 18 170 2550

50 5 30 4 200

3850

Fisher’s Price Index Number = S pnqo ´ S pnqn ´ 100
S poqo S poqn

8.36 Business Mathematics and Statistics

= 3850 ´ 3840 ´ 100 = 1.1 ´ 1.113 ´ 100
3500 3450

= 1.2243 ´ 100 = 1.1064 ´ 100 = 110.64

EXAMPLE 8

Using the data given below calculate the price index number for the year 1988
by (i) Laspeyres’ formula, (ii) Paasche’s formula and (iii) Fisher’s formula with
the year 1979 as base:

Commodity Price (`) Quantity (’000 kg)

Rice 1979 1988 1979 1988
Wheat
Pulses 9.3 4.5 100 90

6.4 3.7 11 10

5.1 2.7 53

Solution: [C.U. B.Com. 1990, 2012, 2014 (G)]
Calculation of Price Index Number

Price (`) Quantity

Commodity 1979 1988 1979 1988 pnqo poqo pnqn poqn
(po) (pn) (qo) (qn)
Rice 9.3 4.5 100 90 450 930 405 837
Wheat 40.7 70.4 37 64
Pulses 6.4 3.7 11 10 13.5 25.5 8.1 15.3
Total 504.2 1025.9 450.1 916.3
5.1 2.7 53

(i) Laspeyres’ Price Index Number = S pnqo ´ 100 = 504.2 ´ 100
S poqo 1025.9

= 0.49147 × 100 = 49.147

(ii) Paasche’s Price Index Number = S pnqn ´ 100 = 450.1 ´ 100
S poqn 916.3

= 0.49121 × 100 = 49.121

(iii) Fisher’s Price Index Number = S pnqo ´ S pnqn ´ 100
S poqo S poqn

= 504.2 ´ 450.1 ´ 100
1025.9 916.3

= 0.49147 ´ 0.49121 ´ 100
= 0.241415 ´ 100
= 0.49134 × 100 = 49.134

Index Numbers 8.37

EXAMPLE 9

Calculate the quantity index for 2015 from the following data using Laspeyres’
formula with 2015 as the base year:

Commodities 2015 2016
Price Quantity Price Quantity
A
B 16 54
C 35 85
48 10 6

[C.U. B.Com. 2016 (G)]

Solution: Calculation of Quantity Index Number

2015 (Base Year) 2016 (Current Year)

Commodities Price Quantity Price Quantity qnpo qopo
(po) (qo) (pn) (qn)
A 4 6
B 16 54 15 15
C 35 85 24 32
Total 48 10 6 43 53

Laspeyres’ Quantity Index Number = S qn po ´ 100
S qo po

= 43 ´ 100
53

= 0.81132 × 100
= 81.132

EXAMPLE 10

Using (i) Laspeyres’ (ii) Paasche’s (iii) Marshall­Edgeworth and (iv) Fisher’s
formula, find the Quantity Index Number from the following data:

Price per unit Quantity (in relevant units)

Commodity Base year Current year Base year Current year

A 4 12 60 50

B 3 10 20 12

C2 6 10 6

[C.U. B.Com. (H) 1992]

8.38 Business Mathematics and Statistics

Solution: Let base year’s price = po
current year’s price = pn
base year’s quantity = qo

current year’s quantity = qn
Calculation of Quantity Index Number

Commodity po pn qo qn qo po qo pn qn po qn pn

A 4 12 60 50 240 720 200 600

B 3 10 20 12 60 200 36 120

C 2 6 10 6 20 60 12 36

Total 320 980 248 756

(i) Laspeyres’ Quantity Index Number = S qn po ´ 100 = 248 ´ 100 = 77.5
S qo po 320

(ii) Paasche’s Quantity Index Number = S qn pn ´100 = 756 ´100 = 77.14
S qo pn 980

(iii) Marshall–Edgeworth Quantity Index Number = S qn ( po + pn ) ´ 100
S qo ( po + pn )

= S qn po + S qn pn ´ 100 = 248 + 756 ´ 100
S qo po + S qo pn 320 + 980

= 1004 ´ 100 = 77.23
1300

(vi) Fisher’s Quantity Index Number = S qn po ´ S qn pn ´ 100
S qo po S qo pn

= 248 ´ 756 ´ 100 = 0.775 ´ 0.771 ´ 100
320 980

= 0.597525 ´ 100 = 0.773 × 100 = 77.3

EXAMPLE 11

Calculate the Price index number for the year 2011 with 2001 as base year using
Laspeyres’ or Paasche’s formula, whichever will be applicable, on the basis of
following data:

Commodity Price (`) Money value (’000 `)
2011
P 2001 2011
Q 240
R 22 30 72
S 150
16 18 36

20 25

8 12

Index Numbers 8.39

(Here money value means total value of commodity which is obtained by
price × quantity)

Solution: Let Po, Pn be the prices for the base year and current year respectively
and qn be the quantity for the current year.

Here, Money value (v) for the current year is given, i.e. pn. qn is given. [v = pn.qn]
So, we can calculate current year’s quantity by using the formula pnqn . But

pn
as the base year’s quantity is not available, we can not use Laspeyres’ formula.
Hence, we have to find the price index number by using Paasche’s formula.

Calculation of Price Index Number

Commodity po pn v = pnqn qn = v poqn
pn
P 22 176
Q 16 30 240 8 64
R 20 120
S 8 18 72 4 24
Total 384
25 150 6

12 36 3

498

Paasche’s Price Index Number = S pnqn ´ 100
S poqn

= 498 ´100
384

= 129.69.

EXAMPLE 12

With the help of the following data, show that Fisher’s formula satisfies the Time
Reversal Test:

Commodities 2011 2012

Rice Quantity (kg) Price (`) Quantity (kg) Price (`)
Wheat
Sugar 50 32 50 30

35 30 40 25
55 16 50 18

[C.U. B.Com. 2013 (G), 2014 (G)]

Solution: Let po, pn be the prices and qo, qn be the quantities for the base year
(2011) and current year (2012) respectively.

8.40 Business Mathematics and Statistics

Calculation of Fisher’s Price Index

Commodities qo po qn pn pnqo poqo pnqn poqn

Rice 50 32 50 30 1500 1600 1500 1600

Wheat 35 30 40 25 875 1050 1000 1200

Sugar 55 16 50 18 990 880 900 800

Total 3365 3530 3400 3600

Fisher’s Price Index of current year with respect to base year

(Pon) = S pnqo ´ S pnqn [omitting the factor 100]
S poqo S poqn

= 3365 ´ 3400
3530 3600

Fisher’s Price Index of base year with respect to current year

(Pno) = S poqn ´ S poqo [omitting the factor 100]
S pnqn S pnqo

= 3600 ´ 3530
3400 3365

Now, Pon × Pno = 3365 ´ 3400 ´ 3600 ´ 3530
3530 3600 3400 3365

= 1 =1

Hence, Fisher’s Price Index satisfies Time Reversal Test.

EXAMPLE 13

Using the following data verify that Paasche’s formula for index does not satisfy
Factor Reversal Test:

Commodity 2005 2008

x Quantity Price (`) Quantity Price (`)
y
z 50 32 50 30
35 30
55 16 40 25

50 18

[C.U. B.Com. 2010, 2016(H)]

Solution: Let po,pn be the prices and qo, qn be the quantities for the base year
(2005) and current year (2008) respectively.

Index Numbers 8.41

Calculation of Paasche’s Index Number

Commodity qo po qn pn poqn poqo pnqn pnqo

x 50 32 50 30 1600 1600 1500 1500

y 35 30 40 25 1200 1050 1000 875

z 55 16 50 18 800 880 900 990

Total 3600 3530 3400 3365

Paasche’s Price Index (Pon) = S pnqn [omitting the factor 100]
S poqn

= 3400
3600

Paasche’s Quantity Index (Qon) = S qn pn [omitting the factor 100]
S qo pn

= 3400
3365

∴ Pon × Qon = 3400 × 3400 = 0.94 × 1.01
3600 3365

= 0.9494

Now, value ratio = S pnqn = 3400 = 0.9632
S poqo 3530

Therefore, Pon × Qon ≠ S pnqn
S poqo

Hence, Paasche’s Index formula does not satisfy Factor Reversal Test.

EXAMPLE 14

Compute Fisher’s index number on the basis of the following data:

Commodity Base year Current year

Price (`) Expenditure (`) Price (`) Expenditure (`)

A 5 25 10 60

B1 10 2 24

C4 16 8 40

D2 40 5 75

Also apply Factor Reversal Test to the above index number.

Solution: We know that, Expenditure = Price × Quantity

Therefore, Quantity = Expenditure
Price

8.42 Business Mathematics and Statistics

Calculation of Fisher’s indices

Base year Current year c e
qn = d
Commodity Price Expenditure Price Expenditure qo = b poqn pnqo
(po) (poqo) (pn) (pnqn)

ab c d e f g hi

A 5 25 10 60 5 6 30 50

B 1 10 2 24 10 12 12 20

C 4 16 8 40 4 5 20 32

D 2 40 5 75 20 15 30 100

Total 91 199 92 202

Fisher’s Price Index Number = S pnqo ´ S pnqn ´ 100
S poqo S poqn

= 202 ´ 199 ´ 100
91 92

= 40198 ´ 100 = 4.8015 ´ 100
8372

= 2.1912 × 100 = 219.12

Fisher’s Quantity Index Number = S poqn ´ S pnqn ´ 100
S poqo S pnqo

= 92 ´ 199 ´ 100 = 18308 ´ 100 = 0.998 × 100 = 99.8
91 202 18382

Now, Pon × Qon = 202 ´ 199 ´ 92 ´ 199 (omitting the factor 100)
91 92 91 202

= 202 ´ 199 ´ 92 ´ 199 = 199 = S pnqn
91 92 91 202 91 S poqo

Hence, Fisher’s Ideal Index Number satisfies the Factor Reversal Test.

EXAMPLE 15

From the following data, calculate the cost of living index numbers:

Group Weight Index numbers
(W) (Base 2004 – 05 = 100)
Food
Clothing 50 241
Fuel etc. 2 221
Rent etc. 3 204
Miscellaneous 16 256
29 179
[C.U. B.Com. 2016 (G)]

Index Numbers 8.43

Solution: Calculation of Cost of Living Index Number

Group Weight Index number I.W
(W) (I)
Food 50 241 12050
Clothing 2 221 442
Fuel etc. 3 204 612
Rent etc. 16 256 4096
Miscellaneous 29 179 5191
100 22391
Total

Cost of Living Index = S IW = 22391 = 223.91
SW 100

EXAMPLE 16

Compare the general cost of living indices for the two years 2006 and 2012 from
the following table:

Group Weight Group Indices

2006 2012

Food 71 370 380
Clothing 3
House Rent 7 423 504
Miscellaneous 10
110 116

279 283

[C.U. B.Com. 2013 (G)]

Solution: Calculation of cost of living indices

Group Weight Group Indices I1W I2W
2006 (I1) 2012 (I2)

Food 71 370 380 26270 26980

Clothing 3 423 504 1269 1512
House Rent 7
110 116 770 812

Miscellaneous 10 279 283 2790 2830

Total 91 31099 32134

Cost of living index for the year 2006 = S I1W = 31099 = 341.75
SW 91

Cost of Living index for the year 2012 = S I2W = 32134 = 353.12
SW 91

8.44 Business Mathematics and Statistics

EXAMPLE 17
Following information relating to the workers in a town are given:

Items Consumer’s Price Index Percentage Expenditure

Food in 2011 (base year = 2000) on the items
Clothing
225 52
Fuel
House Rent 175 8
Miscellaneous
155 10

250 14

150 16

The average wage per month in 2000 was ` 12,000. What should be the

average wage per month in 2011 in that town so that the standard of living in

2011 remains same as that in 2000? [C.U. B.Com. 2013 (H)]

Solution: Calculation of the cost of living index number for the year 2011

Items Consumer’s Price Index in % Expenditure on I.W
2011 (I) (base year = 2000) the items (W)
11700
Food 225 52 1400
1550
Clothing 175 8 3500
2400
Fuel 155 10 20550

House Rent 250 14

Miscellaneous 150 16

Total 100

Cost of living index number for the year 2011 = S IW = 20550 = 205.5
SW 100

Therefore, the average wage per month in 2011 was ` 205.5 with respect to
the base 2000 when the wage per month was ` 100.

Average wage per month

2000 2011
100 205.5
12,000
205.5 ´ 12,000 = `24,660
100

So, to maintain the same standard of living as in 2000, the average wage per
month of the worker should be ` 24,660 in 2011.

Index Numbers 8.45

EXAMPLE 18

Determine the weight for the food group with cost of living index number for
2014 with 2010 as base is 175 from:

Group % increase in expenditure Weight
Food 95 –
Clothing 90 12
Fuel etc. 20 18
Rent etc. 50 20
Miscellaneous 70 10

[C.U. B.Com. 2015(H)]

Solution: Let weight for the food group be ‘x’.

Group % increase in Current Index Weight I.W.
expenditure (I) (W)
Food 195 x 195x
Clothing 95 190 12 2280
Fuel etc. 90 120 18 2160
Rent etc. 20 150 20 3000
Miscellaneous 50 170 10 1700
70 60 + x 9140 + 195x
Total

Cost of living index = S IW
SW

or 175 = 9140 + 195x
60 + x

or 10500 + 175x = 9140 + 195x

or 195 x – 175x = 10500 – 9140

or 20x = 1360

or x = 1360 = 58
20

Therefore, the weight for the food group is 58.

EXAMPLE 19

Consumer price index number goes up from 110 to 200 and the salary of a worker

is also raised from ` 325 to ` 500. Does the worker really gain, and if so, by how

much in real terms? [C.U. B.Com. 1984, 2012]

8.46 Business Mathematics and Statistics

Solution: Consumer price index number goes up from 110 to 200. To maintain
the same standard of living, the salary of the worker should be proportional to
consumer price index number.

Consumer price Salary
index number 325

110

200 325
× 200 = ` 590.91
110

But his salary was raised from ` 325 to ` 500. So, in real terms, he does

not gain. He should get an additional amount of `(590.91 − 500) = ` 90.91 to
maintain same standard of living as before.

EXAMPLE 20

The net salary of an employee was ` 3,000 in the year 2000. The Consumer
Price Index Number in the year 2011 is 250 with 2000 as base year. Calculate the
dearness allowance to be paid to the employee if he has to be rightly compensated.

[C.U. B.Com. 2011]

Solution: Consumer price index number in the year 2011 is 250 with 2000 as
base year. To maintain the same standard of living, salary should be proportional
to consumer price index number.

Consumer price Salary
index number 3000

100

250 3000 × 250 = 7500
100

Therefore, the dearness allowance of ` (7500 − 3000) = ` 4500 to be paid to
the employee if he has to be rightly compensated.

EXAMPLE 21

For the year 2004, the following table gives the cost of living index numbers for
different groups together with their respective weights (1991 as base year):

Group Group index Weight
Food 425 62
Clothes 475 4
Fuel 300 6
House rent 400 12
Misc. 250 16

Index Numbers 8.47

Obtain the overall cost of living index number. Suppose a person was earning

` 6000 in 1991, what should be his earning in 2004 if his standard of living is as

level as in 1991? [C.U. B.Com. 2008]

Solution: Computation of overall cost of living index number

Group Group Index Weight I.W.
(I) (W)
Food 425 62 26350
Clothes 475 4 1900
300 6 1800
Fuel 400 12 4800
House Rent 250 16 4000
100 38850
Misc.
Total

The overall cost of living index number = S IW = 38850 = 388.5
SW 100

The above cost of living index number indicates that a person who is getting
` 100 in 1991 should receive ` 388.5 in 2004 in order to maintain same standard
of living. As per question the person was earning ` 6000 in 1991. Hence, his
earning in 2004 should be as follows:

Cost of living Earnings
Index number
6000
100 6000 × 388.5 = ` 23,310
100
388.5

Therefore, his earning in 2004 should be ` 23,310.

EXAMPLE 22

The following table gives the annual income of a person and the general price
index number for five years:

Year 2010 2011 2012 2013 2014
Income (`) 1800 2100 2500 2750 3000
General Price Index number 100 104 115 160 280

Determine the real income of the person.

Solution:

Real Income = Actual Income ×100
Price Index

8.48 Business Mathematics and Statistics

Calculation of Real Income

Year Income (`) Index number Real income (`)
2010
2011 1800 100 1800 ´ 100 = 1800.00
2012 2100 100
2013 2500
2014 2750 104 2100 ´ 100 = 2019.23
104

115 2500 ´ 100 = 2173.91
115

160 2750 ´ 100 = 1718.75
160

3000 280 3000 ´ 100 = 1071.43
280

EXAMPLE 23

When the cost of an item was increased by 50% a man maintaining his former
scale of consumption said that the rise in price of the item had increased his cost
of living by 5%. What percent of his cost of living was due to buying the item
before the rise of price?

Solution: Let x be the required percentage

Group weight (w) % increase (i) i.w.
Items of increase x 50 50x
Remaining items 0 0
Total 100 − x 50x
100

Increase of cost of living index = 5

Now, 5 = 50x or 50x = 500 or, x = 10.
100

Therefore, 10% of his cost of living was due to buying the item before the rise

of price.

EXAMPLE 24

A price index number series was started in 2006 as base. By 2010 it rose by 25%.

The link relative for 2011 was 95. In this year a new series was started. This new

series rose by 15 points by next year. But during next four years the rise was not

rapid. During 2016 the price level was only 5% higher than 2014 and in 2014 it

was 8% higher than 2012. Splice the two series and calculate the index numbers

for the various years by shifting the base to 2012. [D.U. M.Com. 1983]

Index Numbers 8.49

Solution: Splicing of Index Numbers

Year Index Number Index Number Old Series
(2006 = 100) (2011 = 100) spliced to new

2006 100 – 100 ´ 100 = 84.21
118.75

2010 125 – 125 ´ 100 = 105.26
118.75

2011 æ ´ 95 ö = 118.75 100 100
2012 çè125 100 ÷ø 115 115

–

2014 – æ ´ 108 ö 124.2 124.20
çè115 100 ø÷

2016 – æ ´ 105 ö 130.41 130.41
èç124.2 100 ÷ø

Shifting base to 2012 Index Number
Year 84.21´ 100 = 73.23
2006
2010 115
2011 105.26 ´ 100 = 91.53
2012
2014 115
2016 100 ´ 100 = 86.96

115
115´ 100 = 100.00

115
124.20 ´ 100 = 108.00

115
130.41´ 100 = 113.40

115

EXAMPLE 25

From the data given below, construct a cost of living index number by using
family budget method for 2016 with 2006 an base year:

Commodity A BCDE F
Qty in units in 2006 50 25 10 20 30 40
Price per unit in 2006 (`): 10 58796
Price per unit in 2016 (`): 6 4 3 8 10 12

[Osmania University, B.Com. 1986]

8.50 Business Mathematics and Statistics

Solution: Calculation of cost of living index by using family budget method

Commodity Qty in Price in Price in I = Pn ´ 100 W = poqo I.W.
2006 (qo) 2006 (po) 2016 (pn) Po

A 50 10 6 60.00 500 30,000.00

B 25 5 4 80.00 125 10,000.00

C 10 8 3 37.50 80 3,000.00

D 20 7 8 114.28 140 15,999.20

E 30 9 10 111.11 270 29,999.70

F 40 6 12 200.00 240 48,000.00

Total 1355 1,36,998.90

EXAMPLE 26

Construct chain index numbers (Base 1992 = 100) for the year 1993–97.

Year 1993 1994 1995 1996 1997

Link Index 103 98 105 112 108

Solution: [C.U. B.Com. 1999]
Calculation of chain index numbers

Year Link Index Chain Index Number
1992 100 100

1993 103 ´ 100
103 = 103

100

1994 98 ´103
98 = 100.94

100

1995 105 ´ 100.94
105 = 105.987

100

1996 112 112 ´105.987 = 118.71
100

1997 108 ´118.71
108 = 128.21

100

EXAMPLE 27

In the following series of index numbers, shift the base from 1970 to 1973.

Year 1970 1971 1972 1973 1974 1975 1976 1977
Index 100 105 110 125 135 180 195 205

[Lucknow University, 1982]